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Functional spaces associated with the Weyl-Hörmander calculus. (Espaces fonctionnels associés au calcul de Weyl-Hörmander.) (French) Zbl 0798.35172
Following an idea of R. Beals [Duke Math. J. 42, 1-42 (1975; Zbl 0343.35078)], the authors consider Sobolev spaces associated to general classes of pseudo-differential operators. Precisely, if \(g\) is a metric of Hörmander and \(M\) a related weight function, \(H(M,g)\) is defined as the space of all \(u\in {\mathcal S}'(\mathbb{R}^ n)\) such that \(a^ W(x,D)u\in L^ 2(\mathbb{R}^ n)\) for all Weyl pseudo-differential operators with symbol \(a(x,\eta)\in S(M,g)\). An equivalent definition of \(H(M,g)\) is given by the authors in terms of decompositions of Littlewood-Paley type. Such decompositions allow in turn an abstract characterization of the class of the operators with symbol \(a(x,\eta)\) in \(S(M,g)\) and applications to the so-called problem of the spectral invariance. Similar results for the classes \(S^ m_{\rho,\delta}\) were obtained by J. Ueberberg [Manuscr. Math. 61, No. 4, 459-375 (1988; Zbl 0674.47033)].
Reviewer: L.Rodino (Torino)

MSC:
35S05 Pseudodifferential operators as generalizations of partial differential operators
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47G30 Pseudodifferential operators
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References:
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